By the "Angle-Side-Angle" theorem from elementary geometry,...

By the "Angle-Side-Angle" theorem from elementary geometry, triangles $C D O$ and $A O B$ in the figure are congruent. Explain how this proves that if $B$ has coordinates $(x, y),$ then $D$ has coordinates $(-y, x) .$ Then explain how the figure shows that the following reduction formulas are valid: $$\begin{array}{l}{\sin \left(t+\frac{\pi}{2}\right)=\cos t} \\ {\cos \left(t+\frac{\pi}{2}\right)=-\sin t} \\ {\tan \left(t+\frac{\pi}{2}\right)=-\cot t}\end{array}$$

By the "Angle-Side-Angle" theorem from elementary geometry, triangles $C D O$ and $A O B$ in the figure are congruent. Explain how this proves that if $B$ has coordinates $(x, y),$ then $D$ has coordinates $(-y, x) .$ Then explain how the figure shows that the following reduction formulas are valid:
$$\begin{array}{l}{\sin \left(t+\frac{\pi}{2}\right)=\cos t} \\ {\cos \left(t+\frac{\pi}{2}\right)=-\sin t} \\ {\tan \left(t+\frac{\pi}{2}\right)=-\cot t}\end{array}$$

Key Concepts

Angle-Side-Angle (ASA) Congruence Theorem

This theorem establishes that if two angles and the included side of one triangle are respectively equal to two angles and the included side of another triangle, then the triangles are congruent. In the context of the problem, congruence is used to show that rotating one triangle results in another triangle with identical shape and dimensions, which directly supports the coordinate transformation of points under rotation.

Rotation Transformation in the Plane

A rotation transformation in the plane, especially a 90° (?/2 radians) rotation, has the property of mapping any point (x, y) to (-y, x). This concept is fundamental in explaining how the point with coordinates (x, y) is transformed to (-y, x) and how this rotation is connected to the trigonometric reduction formulas through the geometric changes in the coordinate positions.

Trigonometric Reduction Formulas

Reduction formulas in trigonometry, such as sin(t + ?/2) = cos t, cos(t + ?/2) = -sin t, and tan(t + ?/2) = -cot t, simplify the evaluation of trigonometric functions at shifted angles by expressing them in terms of functions at a standard angle. These formulas naturally emerge from the coordinate transformations associated with rotating points in the plane, as demonstrated in the diagram.

Unit Circle and Trigonometric Coordinates

The unit circle provides a geometric framework where the cosine and sine of an angle are interpreted as the x and y coordinates, respectively, of a point on the circle. A rotation by ?/2 swaps these coordinates according to a specific pattern, which is the geometric underpinning of why and how the trigonometric functions change when their arguments are increased by ?/2 radians.

Transcript

00:01 So now we're looking at a picture and we're looking at this angle being t and it crosses the unit circle at x, y.

00:10 And then we're going to rotate this.

00:13 And so let me just draw this down.

00:16 We know that this length right here is x.

00:19 And because it's in the first quadrant the way i've drawn it, it would be positive.

00:22 And this is positive y if we're in the first t was in the first quadrant.

00:26 Now let's rotate it so that it's going to come over here and rotate pi over two.

00:35 And then this coordinate right here is what we want to figure out what those values are.

00:42 We are kind of told it in the diagram and the problem, but we're going to figure it out.

00:46 So if we take this triangle and rotate it 90 degrees, notice that this length will end up moving right here.

00:54 So if i rotate the triangle 90 degrees, this little segment here, which is why, is going to move here.

01:02 However, in the second quadrant, this would actually be a negative value.

01:07 So this coordinate would be the opposite of whatever that coordinate was.

01:14 So the x coordinate on the unit circle would be the opposite of this value for that angle.

01:21 And then this length, if we rotate it, 90 degrees, would be this length.

01:27 It would be from here to here.

01:30 So this x value would be this x value.

01:32 Well, that's because this would form a rectangle the way i drew it.

01:37 This length would be x.

01:39 And it would be the same value.

01:40 If this is positive, this would also be positive because it's going up.

01:45 So the y coordinate for this point is actually x...

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